Name: Magnetic Susceptibility and Relative Magnetic Permeability
Uploaded: 2019-07-19
Duration: 7 min 13 s
Description: Magnetic susceptibility and relative magnetic permeability sound like complicated terms, but they're not. In this video, we will learn what they are and the...

Question

Explain the terms magnetisation, magnetising field intensity, permeability relative permeability and magnetic susceptibility Write the relation between permeability and susceptibility.

Accepted Answer

(a) Magnetic Induction $(\vec{B})$ : When a piece of any substance is placed in an external magnetic field, the substance becomes magnetised. The magnetism so produced in the substance is called ‘induced magnetism’ and this phenomenon is called ‘magnetic induction’.

An iron bar is placed in a uniform magnetic field with its length parallel to the magnetic lines of force. The bar is magnetised by induction, with a south pole induced on the left end where lines of force enter the bar and a north pole induced on the right end where lines of force leave the bar.

The magnetised bar produces its own magnetic field. Its lines of force (shown by broken curves) are in the same direction as those of the original field inside the bar but in opposite direction outside the bar. This results in a concentration of the lines of force within the bar The magnetic flux density within the bar is increased whereas it becomes quite weak at certain places outside the bar.
The magnetic lines of force inside the magnetised bar are called ‘magnetic lines of induction’.

The number of magnetic lines of induction inside a magnetised substance crossing unit area normal to their direction is called the magnitude of magnetic induction or magnetic flux density, inside the substance. Magnetic induction is a vector, denoted by $\vec{B,}$ whose direction at any point is the direction of magnetic line of induction at that point.

The $SI$ unit of magnetic induction is Tesla $(T)$ or $weber / {meter}^{2} (Wb m^{- 2})$ or $newton / (ampere - metre) (N A^{- 1} m^{- 1}) .$

The $CGS$ unit of $\vec{B}$ is ‘gauss’ (G), where $1 G = 10^{- 4} T$

(b) Intensity of Magnetisation (M) : The intensity of magnetisation, or simply magnetisation of a magnetised substance represents the extent to which the substance is magnetised. It is defined as the magnetic moment per unit volume of the magnetised substance and is denoted by $\vec{M}$ . Numerically, $\vec{M} = \frac{\vec{m}}{V}$

where $\vec{m}$ is the magnetic moment of a magnetised substance of volume $V .$ The $SI$ unit of magnetisation is $ampere / metre ({Am}^{- 1}) .$

(c) Magnetic Intensity or Magnetic Field Strength $(\vec{H})$ : When a substance is placed in an external magnetic field, it becomes magnetised. The actual magnetic field inside the substance is the sum of the external field and the field due to its magnetisation. The capability of the magnetising field to magnetise the substance is expressed by means of a vector $\vec{H},$ called the ‘magnetic intensity’ of the field. The magnetic intensity $\vec{H}$ is defined through the vector relation

$\vec{H} = \frac{\vec{B}}{μ_{0}} - \vec{M}$

where $\vec{B}$ is magnetic field induction inside the substance and $\vec{M}$ is the intensity of magnetisation. $μ_{0}$ is permeability of free space.

From practical point of view, the $\vec{H}$ field is very important, especially in dealing with ferromagnetic substances. It can be controlled by adjusting current, say in coils or solenoids.
The SI unit of $\vec{H}$ is same as of $\vec{M}$ , that is, ${Am}^{- 1}$

The $CGS$ unit of $\vec{H}$ is ‘oersted’. In view of the above vector relation, we can write

$1 oersted = 1 gauss / μ_{0} = 10^{- 4} T / μ_{0}$

$But μ_{0} = 4 π \times 10^{- 7} T m A^{- 1}$

$∴ 1 oersted = \frac{10^{- 4} T}{4 π \times 10^{- 7} T {mA}^{- 1}} ≃ 80 {Am}^{- 1}$

This is the relation of the oersted to the $SI$ unit.

(d) Magnetic Permeability (p): The magnetic permeability of a substance is a measure of its conduction of magnetic lines of force through it. It is defined as the ratio of the magnetic induction $\vec{B}$ inside the magnetised substance to the magnetic intensity $\vec{H}$ of the magnetising field, that is, $μ = \frac{\vec{B}}{\vec{H}}$

Numerically, $μ = B / H$

The $SI$ unit of permeability is $newton / {ampere}^{2} Z (NA^{- 2})$ , or $tesla - metre / ampere (TmA^{- 1})$ , or

$weber / ampere - metre (Wb A^{- 1} m^{- 1}) .$

(e) Relative Magnetic Permeability $(μ_{r})$ : The relative magnetic permeability of a substance is the ratio of the magnetic permeability $μ$ of the substance to the permeability of free space $μ_{0}$ that is, $μ_{r} = \frac{μ}{μ_{0}}$

It is a dimensionless quantity and is equal to $1$ for vacuum (by definition).

Alternatively, the relative permeability of a substance is defined as the ratio of the magnetic flux density $B$ in the substance when placed in a magnetic field and the flux density $B_{0}$ in vacuum in the same field, that is, $μ_{r} = \frac{B}{B_{0}}$

(f) Magnetic Susceptibility $(χ_{m})$ : It is a measure of how easily a substance is magnetised in a magnetising field. For paramagnetic and diamagnetic substances, the magnetisation $\vec{M}$ is directly proportional to the magnetic intensity $\vec{H}$ of the magnetising field. That is, $\vec{M} = χ_{m} \vec{H}$

The constant $χ_{m}$ is called the ‘magnetic susceptibility’ of the substance. It may be defined as the ratio of the intensity of magnetisation to the magnetic intensity of the magnetising field, that is, $χ_{m} = \frac{M}{H}$

$X_{m}$ is dimensionless (a pure number) because $M$ and $H$ have same unit. Its value for vacuum is zero as there can be no magnetisation in vacuum.

We can classify substances in terms of $x_{m}$

Substances having (small) negative values $(< 0)$ $x_{m}$ are diamagnetic (as $\vec{M}$ is opposite $\vec{H}$ ); those having small positive values of xm (> 0) are paramagnetic; and those having very large positive values of $X_{m} (> > 0)$ are ferromagnetic. However, for ferromagnetic substances, $\vec{M}$ is not accurately proportional to $\vec{H,}$ and so $X_{m}$ is not strictly constant.

Relation between Relative Permeability $(m_{r})$ and Magnetic Susceptibility $(X_{m})$ : When a substance is placed in a magnetising field, it becomes magnetised. The total magnetic flux density B within the substance is the flux density that would have been produced by the magnetising field in vacuum plus the flux density due to the magnetisation of the substance. If M be the intensity of magnetisation of the substance, then, by definition, the magnetic intensity of the magnetising field is given by

$H = \frac{B}{μ_{0}} - M or B = μ_{0} (H + M)$

But $M = χ_{m} H$ where $χ_{m}$ is the magnetic susceptibility of the substance.

$∴ B = μ_{0} (1 + χ_{m}) H$

Again, $B = μ H$ , where $μ$ is the permeability of the substance.

$∴ μ = μ_{0} (1 + χ_{m})$

Now $\frac{μ}{μ_{0}}$ is the relative permeability $μ_{r}$

$∴ μ_{r} = 1 + χ_{m}$

$μ_{r}$ is slightly less than $1$ for diamagnetic substances; slightly more than $1$ for paramagnetic substances; and very large than $1$ for ferromagnetic substances.

Explain the terms magnetisation, magnetising field intensity, permeability relative permeability and magnetic susceptibility Write the relation between permeability and susceptibility.

Important Questions on Magnetic Classification of Substances

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